Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
  1. Home
  2. Books
  3. Fractals in Probability and Analysis
  • Cited by 49
Publisher:
Cambridge University Press
Online publication date:
January 2017
Print publication year:
2016
Online ISBN:
9781316460238
DOI:
https://doi.org/10.1017/9781316460238
Subjects:
Mathematics (general), Mathematics, Real and Complex Analysis, Probability Theory and Stochastic Processes, Statistics and Probability
Series:
Cambridge Studies in Advanced Mathematics (162)
Information

Book description

This is a mathematically rigorous introduction to fractals which emphasizes examples and fundamental ideas. Building up from basic techniques of geometric measure theory and probability, central topics such as Hausdorff dimension, self-similar sets and Brownian motion are introduced, as are more specialized topics, including Kakeya sets, capacity, percolation on trees and the traveling salesman theorem. The broad range of techniques presented enables key ideas to be highlighted, without the distraction of excessive technicalities. The authors incorporate some novel proofs which are simpler than those available elsewhere. Where possible, chapters are designed to be read independently so the book can be used to teach a variety of courses, with the clear structure offering students an accessible route into the topic.

Reviews

'Fractal sets are now a key ingredient of much of mathematics, ranging from dynamical systems, transformation groups, stochastic processes, to modern analysis. This delightful book gives a correspondingly broad view of fractal sets. The presentation is original, clear and thoughtful, often with new and interesting approaches. It is suited both to graduate students and researchers, discussing reasonably easily accessible questions as well as research topics that are being actively investigated today. For example, in addition to learning about fractals, students will get new insights into some core topics, such as Brownian motion, while researchers will find new ideas for up-to-date research, for example related to analysts’ traveling salesman problems. The book is splendid for a variety of graduate courses, most sections being essentially independent of each other, and is supported by a very large number of exercises of varying levels with hints and solutions.'

Pertti Mattila - University of Helsinki

'This is a wonderful book, introducing the reader into the modern theory of fractals. It uses tools from analysis and probability very elegantly, and starting from the basics ends with a selection of deep and important results. The authors worked hard to achieve clarity; the book contains many original proofs which are expository gems. The book would serve very well for a graduate course; it is highly recommended both for students and for experts. A notable feature is a wide selection of exercises, some quite challenging, but made more accessible with an appendix containing selected hints.'

Boris Solomyak - Bar-Ilan University, Israel

'This is a very valuable contribution to the field of geometric measure theory and its interactions with other branches of mathematical analysis and probability. The notions of Hausdorff measure, Hausdorff dimension and Minkowski dimension are central objects in this text, as in other books on geometric measure theory. What is special in this text, written by two major experts in geometric analysis and probability, is the emphasis on problems lying in the intersection of probability and analysis. In particular, the book studies a variety of questions in connection with self-similar sets, Frostman's theory, Weierstrass functions, Brownian motion and its relationship with the Dirichlet problem for harmonic functions, Besicovitch–Kakeya sets, and Jones' traveling salesman theorem. Many of the problems considered in the book are difficult to find in the literature. Further, very often their proofs are based on new and illuminating arguments. All in all, I think that this is a great book.'

Xavier Tolsa - ICREA, Catalan Institution for Research and Advanced Studies, and Universitat Autònoma de Barcelona

'This book, written by two of the best specialists in the world, is centered on the probabilistic aspects of geometric measure theory and fractals, but also contains beautiful pure analysis arguments. The point of view is very concrete, often based on many interesting examples or methods rather than a general theory. The most impressive aspect of the book is the huge collection of exercises of all levels, which will make a serious reading of the book both a pleasure and, if the reader wants to do them all, a performance.'

Guy David - Université de Paris Sud

'There are at least two outstanding features of Bishop-Peres’s new textbook that will help it stand with self-assurance … The first feature is the remarkable clarity of exposition. The proofs are beautifully presented, with a stress on communicating ideas and methods (over technicalities). This leads the authors to study the simplest cases of problems/results that already contain the most important ideas. The second feature, which moves this text into its own class among existing graduate texts on the subject, is an exceptional list of 378 exercises.'

Tushar Das Source: MAA Reviews

'This is a technical monograph suited to practioners of geometric measure theory and analysis written by two of the world’s leaders in the field. It would make a serious study for graduate students, containing a large number of helpful examples.'

Chris Athorne Source: Contemporary Physics

Refine List

Actions for selected content:

Select all | Deselect all
  • View selected items
  • Export citations
  • Download PDF (zip)
  • Save to Kindle
  • Save to Dropbox
  • Save to Google Drive

Save Search

You can save your searches here and later view and run them again in "My saved searches".

Please provide a title, maximum of 40 characters.
Cancel
×

Contents

Contents
References
References
Adelman, Omer. 1985. Brownian motion never increases: a new proof to a result of Dvoretzky, Erdʺos and Kakutani. Israel J. Math., 50(3), 189–192.
Adelman, Omer, Burdzy, Krzysztof, and Pemantle, Robin. 1998. Sets avoided by Brownian motion. Ann. Probab., 26(2), 429–464.
Aizenman, M., and Burchard, A. 1999. Hölder regularity and dimension bounds for random curves. Duke Math. J., 99(3), 419–453.
Ala-Mattila, Vesa. 2011. Geometric characterizations for Patterson–Sullivan measures of geometrically finite Kleinian groups. Ann. Acad. Sci. Fenn. Math. Diss., 120. Dissertation, University of Helsinki, Helsinki, 2011.
Arora, Sanjeev. 1998. Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. J. ACM, 45(5), 753–782.
Arveson, William. 1976. An Invitation to C*-algebras. Springer-Verlag, New York–Heidelberg. Graduate Texts in Mathematics, No. 39.
Athreya Krishna, B., and Ney Peter, E. 1972. Branching Processes. New York: Springer-Verlag. Die Grundlehren der Mathematischen Wissenschaften, Band 196.
Avila, Artur, and Lyubich, Mikhail. 2008. Hausdorff dimension and conformal measures of Feigenbaum Julia sets. J. Amer. Math. Soc., 21(2), 305–363.
Azzam, Jonas. 2015. Hausdorff dimension of wiggly metric spaces. Ark. Mat., 53(1), 1–36.
Babichenko, Yakov, Peres, Yuval, Peretz, Ron, Sousi, Perla, and Winkler, Peter. 2014. Hunter, Cauchy rabbit, and optimal Kakeya sets. Trans. Amer. Math. Soc., 366(10), 5567–5586.
Bachelier, L. 1900. Théorie de la speculation. Ann. Sci. Ecole Norm. Sup., 17, 21–86.
Balka, Richárd, and Peres, Yuval. 2014. Restrictions of Brownian motion. C. R. Math. Acad. Sci. Paris, 352(12), 1057–1061.
Balka, Richárd, and Peres, Yuval. 2016. Uniform dimension results for fractional Brownian motion. preprint, arXiv:1509.02979 [math.PR].
Bandt, Christoph, and Graf, Siegfried. 1992. Self-similar sets. VII. A characterization of self-similar fractals with positive Hausdorff measure. Proc. Amer. Math. Soc., 114(4), 995–1001.
Barański, Krzysztof. 2007. Hausdorff dimension of the limit sets of some planar geometric constructions. Adv. Math., 210(1), 215–245.
Barański, Krzysztof, Bárány, Balázs, and Romanowska, Julia. 2014. On the dimension of the graph of the classical Weierstrass function. Adv. Math., 265, 32–59.
Barlow Martin, T., and Perkins, Edwin. 1984. Levels at which every Brownian excursion is exceptional. Pages 1–28 of: Seminar on probability, XVIII. Lecture Notes in Math., vol. 1059. Berlin: Springer.
Barlow Martin, T., and Taylor S., James. 1992. Defining fractal subsets of Z d . Proc. London Math. Soc. (3), 64(1), 125–152.
Bass Richard, F. 1995. Probabilistic Techniques in Analysis. Probability and its Applications (New York). New York: Springer-Verlag.
Bass Richard, F., and Burdzy, Krzysztof. 1999. Cutting Brownian paths. Mem. Amer. Math. Soc., 137(657), x+95.
Bateman, Michael, and Katz Nets, Hawk. 2008. Kakeya sets in Cantor directions. Math. Res. Lett., 15(1), 73–81.
Bateman, Michael, and Volberg, Alexander. 2010. An estimate from below for the Buffon needle probability of the four-corner Cantor set. Math. Res. Lett., 17(5), 959–967.
Benjamini, Itai, and Peres, Yuval. 1992. Random walks on a tree and capacity in the interval. Ann. Inst. H. Poincaré Probab. Statist., 28(4), 557–592.
Benjamini, Itai, and Peres, Yuval. 1994. Tree-indexed random walks on groups and first passage percolation. Probab. Theory Related Fields, 98(1), 91–112.
Benjamini, Itai, Pemantle, Robin, and Peres, Yuval. 1995. Martin capacity for Markov chains. Ann. Probab., 23(3), 1332–1346.
Berman Simeon, M. 1983. Nonincrease almost everywhere of certain measurable functions with applications to stochastic processes. Proc. Amer. Math. Soc., 88(1), 141–144.
Bertoin, Jean. 1996. Lévy Processes. Cambridge Tracts in Mathematics, vol. 121. Cambridge: Cambridge University Press.
Bertrand-Mathis, Anne. 1986. Ensembles intersectifs et récurrence de Poincaré. Israel J. Math., 55(2), 184–198.
Besicovitch, A.S. 1919. Sur deux questions d'intégrabilité des fonctions. J. Soc. Phys.- Math. (Perm'), 2(1), 105–123.
Besicovitch, A.S. 1928. On Kakeya's problem and a similar one. Math. Z., 27(1), 312–320.
Besicovitch, A.S. 1935. On the sum of digits of real numbers represented in the dyadic system. Math. Ann., 110(1), 321–330.
Besicovitch, A.S. 1938a. On the fundamental geometrical properties of linearly measurable plane sets of points (II). Math. Ann., 115(1), 296–329.
Besicovitch, A.S. 1938b. On the fundamental geometrical properties of linearly measurable plane sets of points II. Math. Ann., 115, 296–329.
Besicovitch, A.S. 1952. On existence of subsets of finite measure of sets of infinite measure. Nederl. Akad. Wetensch. Proc. Ser. A. 55 = Indagationes Math., 14, 339–344.
Besicovitch, A.S. 1956. On the definition of tangents to sets of infinite linear measure. Proc. Cambridge Philos. Soc., 52, 20–29.
Besicovitch, A.S. 1964. On fundamental geometric properties of plane line-sets. J. London Math. Soc., 39, 441–448.
Besicovitch, A.S., and Moran, P.A.P. 1945. The measure of product and cylinder sets. J. London Math. Soc., 20, 110–120.
Besicovitch, A.S., and Taylor, S.J. 1954. On the complementary intervals of a linear closed set of zero Lebesgue measure. J. London Math. Soc., 29, 449–459.
Besicovitch, A.S., and Ursell, H.D. 1937. Sets of fractional dimension v: On dimensional numbers of some continuous curves. J. London Math. Soc., 12, 18–25.
Bickel Peter, J. 1967. Some contributions to the theory of order statistics. Pages 575– 591 of: Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calf., 1965/66), Vol. I: Statistics. Berkeley, Calif.: Univ. California Press.
Billingsley, Patrick. 1961. Hausdorff dimension in probability theory. II. Illinois J. Math., 5, 291–298.
Binder, Ilia, and Braverman, Mark. 2009. The complexity of simulating Brownian motion. Pages 58–67 of: Proceedings of the Twentieth Annual ACM–SIAM Symposium on Discrete Algorithms. Philadelphia, PA: SIAM.
Binder, Ilia, and Braverman, Mark. 2012. The rate of convergence of the walk on spheres algorithm. Geom. Funct. Anal., 22(3), 558–587.
Bishop Christopher, J 1996. Minkowski dimension and the Poincaré exponent. Michigan Math. J., 43(2), 231–246.
Bishop Christopher, J 1997. Geometric exponents and Kleinian groups. Invent. Math., 127(1), 33–50.
Bishop Christopher, J 2001. Divergence groups have the Bowen property. Ann. of Math. (2), 154(1), 205–217.
Bishop Christopher, J 2002. Non-rectifiable limit sets of dimension one. Rev. Mat. Iberoamericana, 18(3), 653–684.
Bishop Christopher, J 2007. Conformal welding and Koebe's theorem. Ann. of Math. (2), 166(3), 613–656.
Bishop Christopher, J, and Jones Peter, W. 1990. Harmonic measure and arclength. Ann. of Math. (2), 132(3), 511–547.
Bishop Christopher, J, and Jones Peter, W. 1994a. Harmonic measure, L2 estimates and the Schwarzian derivative. J. Anal. Math., 62, 77–113.
Bishop Christopher, J, and Jones Peter, W. 1994b. Harmonic measure, L2 estimates and the Schwarzian derivative. J. D'Analyse Math., 62, 77–114.
Bishop Christopher, J, and Jones Peter, W. 1997. Hausdorff dimension and Kleinian groups. Acta Math., 179(1), 1–39.
Bishop Christopher, J, and Peres, Yuval. 1996. Packing dimension and Cartesian products. Trans. Amer. Math. Soc., 348(11), 4433–4445.
Bishop Christopher, J, and Steger, Tim. 1993. Representation-theoretic rigidity in PSL(2, R). Acta Math., 170(1), 121–149.
Bishop Christopher, J, Jones Peter, W., Pemantle, Robin, and Peres, Yuval. 1997. The dimension of the Brownian frontier is greater than 1. J. Funct. Anal., 143(2), 309–336.
Bonk, Mario. 2011. Uniformization of Sierpiński carpets in the plane. Invent. Math., 186(3), 559–665.
Bonk, Mario, and Merenkov, Sergei. 2013. Quasisymmetric rigidity of square Sierpiński carpets. Ann. of Math. (2), 177(2), 591–643.
Bonk, Mario, Kleiner, Bruce, and Merenkov, Sergei. 2009. Rigidity of Schottky sets. Amer. J. Math., 131(2), 409–443.
Bourgain, J. 1987. Ruzsa's problem on sets of recurrence. Israel J. Math., 59(2), 150– 166.
Bourgain, J. 1991. Besicovitch type maximal operators and applications to Fourier analysis. Geom. Funct. Anal., 1(2), 147–187.
Bourgain, J. 1999. On the dimension of Kakeya sets and related maximal inequalities. Geom. Funct. Anal., 9(2), 256–282.
Boyd David, W. 1973. The residual set dimension of the Apollonian packing. Mathematika, 20, 170–174.
Brown, R. 1828. A brief description of microscopical observations made in the months of June, July and August 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies. Ann. Phys., 14, 294–313.
Burdzy, Krzysztof. 1989. Cut points on Brownian paths. Ann. Probab., 17(3), 1012– 1036.
Burdzy, Krzysztof. 1990. On nonincrease of Brownian motion. Ann. Probab., 18(3), 978–980.
Burdzy, Krzysztof, and Lawler Gregory, F. 1990. Nonintersection exponents for Brownian paths. II. Estimates and applications to a random fractal. Ann. Probab., 18(3), 981–1009.
Cajar, Helmut. 1981. Billingsley Dimension in Probability Spaces. Lecture Notes in Mathematics, vol. 892. Berlin: Springer-Verlag.
Carleson, Lennart. 1967. Selected Problems on Exceptional Sets. Van Nostrand Mathematical Studies, No. 13. D. Van Nostrand Co., Inc., Princeton N.J.–Toronto, Ont.–London.
Carleson, Lennart. 1980. The work of Charles Fefferman. Pages 53–56 of: Proceedings of the International Congress of Mathematicians (Helsinki, 1978). Helsinki: Acad. Sci. Fennica.
Chang, S.-Y.A., Wilson, J.M., and Wolff, T.H. 1985. Some weighted norm inequalities concerning the Schrödinger operators. Comment. Math. Helv., 60(2), 217–246.
Charmoy Philippe, H.A., Peres, Yuval, and Sousi, Perla. 2014. Minkowski dimension of Brownian motion with drift. J. Fractal Geom., 1(2), 153–176.
Chayes, J.T., Chayes, L., and Durrett, R. 1988. Connectivity properties of Mandelbrot's percolation process. Probab. Theory Related Fields, 77(3), 307–324.
Cheeger, Jeff, and Kleiner, Bruce. 2009. Differentiability of Lipschitz maps from metric measure spaces to Banach spaces with the Radon-Nikodým property. Geom. Funct. Anal., 19(4), 1017–1028.
Chow Yuan, Shih, and Teicher, Henry. 1997. Probability Theory. Third edn. Independence, Interchangeability, Martingales. Springer Texts in Statistics. New York: Springer-Verlag.
Christ, Michael. 1984. Estimates for th. k-plane transform. Indiana Univ. Math. J., 33(6), 891–910.
Christensen Jens Peter, Reus. 1972. On sets of Haar measure zero in abelian Polish groups. Pages 255–260 (1973) of: Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces (Jerusalem, 1972), vol. 13.
Ciesielski, Z., and Taylor, S.J. 1962. First passage times and sojourn times for Brownian motion in space and the exact Hausdorff measure of the sample path. Trans. Amer. Math. Soc., 103, 434–450.
Colebrook, C.M. 1970. The Hausdorff dimension of certain sets of nonnormal numbers. Michigan Math. J., 17, 103–116.
Cόrdoba, Antonio. 1993. The fat needle problem. Bull. London Math. Soc., 25(1), 81–82.
Cover Thomas, M., and Thomas Joy, A. 1991. Elements of Information Theory. Wiley Series in Telecommunications. New York: John Wiley & Sons Inc.
Cunningham, Jr., F. 1971. The Kakeya problem for simply connected and for starshaped sets. Amer. Math. Monthly, 78, 114–129.
Cunningham, Jr., F. 1974. Three Kakeya problems. Amer. Math. Monthly, 81, 582–592.
David, Guy. 1984. Opérateurs intégraux singuliers sur certaines courbes du plan complexe. Ann. Sci. École Norm Sup., 17, 157–189.
David, Guy. 1998. Unrectifiable 1-sets have vanishing analytic capacity. Rev. Mat. Iberoamericana, 14(2), 369–479.
Davies Roy, O. 1952. On accessibility of plane sets and differentiation of functions of two real variables. Proc. Cambridge Philos. Soc., 48, 215–232.
Davies Roy, O. 1971. Some remarks on the Kakeya problem. Proc. Cambridge Philos. Soc., 69, 417–421.
de Leeuw, Karel. 1965. O. Lp multipliers. Ann. of Math. (2), 81, 364–379.
Dekking, F.M., and Grimmett, G.R. 1988. Superbranching processes and projections of random Cantor sets. Probab. Theory Related Fields, 78(3), 335–355.
Drury, S.W. 1983. Lp estimates for the X-ray transform. Illinois J. Math., 27(1), 125–129.
Dubins Lester, E. 1968. On a theorem of Skorohod. Ann. Math. Statist., 39, 2094–2097.
Dudley, R.M. 2002. Real Analysis and Probability. Cambridge Studies in Advanced Mathematics, vol. 74. Cambridge University Press, Cambridge. Revised reprint of the 1989 original.
Dudziak James, J. 2010. Vitushkin's Conjecture for Removable Sets. Universitext. New York: Springer.
Duistermaat, J.J. 1991. Self-similarity of “Riemann's nondifferentiable function”. Nieuw Arch. Wisk. (4), 9(3), 303–337.
Duplantier, Bertrand. 2006. Brownian motion, “diverse and undulating”. Pages 201– 293 of: Einstein, 1905–2005. Prog. Math. Phys., vol. 47. Basel: Birkhäuser. Translated from the French by Emily Parks.
Durrett, Richard. 1996. Probability: Theory and Examples. Belmont, CA: Duxbury Press.
Dvir, Zeev. 2009. On the size of Kakeya sets in finite fields. J. Amer. Math. Soc., 22(4), 1093–1097.
Dvir, Zeev, and Wigderson, Avi. 2011. Kakeya sets, new mergers, and old extractors. SIAM J. Comput., 40(3), 778–792.
Dvir, Zeev, Kopparty, Swastik, Saraf, Shubhangi, and Sudan, Madhu. 2009. Extensions to the method of multiplicities, with applications to Kakeya sets and mergers. Pages 181–190 of: 2009 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2009). IEEE Computer Soc., Los Alamitos, CA.
Dvoretzky, A., Erdʺos, P., and Kakutani, S. 1950. Double points of paths of Brownian motion i. n-space. Acta Sci. Math. Szeged, 12(Leopoldo Fejer et Frederico Riesz LXX annos natis dedicatus, Pars B), 75–81.
Dvoretzky, A., Erdʺos, P., and Kakutani, S. 1961. Nonincrease everywhere of the Brownian motion process. Pages 103–116 of: Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. II. Berkeley, Calif.: Univ. California Press.
Dynkin, E.B., and Yushkevich, A.A. 1956. Strong Markov processes. Theory Probab. Appl., 1, 134–139.
Edgar Gerald, A. (ed). 2004. Classics on Fractals. Studies in Nonlinearity. Westview Press. Advanced Book Program, Boulder, CO.
Edmonds, Jack. 1965. Paths, trees, and flowers. Canad. J. Math., 17, 449–467.
Eggleston, H.G. 1949. The fractional dimension of a set defined by decimal properties. Quart. J. Math., Oxford Ser., 20, 31–36.
Einstein, A. 1905. Über die von der molekularkinetischen Theorie der Warme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Ann. Physik, 17, 549–560.
Elekes, György, Kaplan, Haim, and Sharir, Micha. 2011. On lines, joints, and incidences in three dimensions. J. Combin. Theory Ser. A, 118(3), 962–977.
Elekes, Márton, and Steprāns, Juris. 2014. Haar null sets and the consistent reflection of non-meagreness. Canad. J. Math., 66(2), 303–322.
Elekes, Márton, and Vidnyánszky, Zoltán. 2015. Haar null sets without G™?hulls. Israel J. Math., 209(1), 199–214.
Ellenberg Jordan, S., Oberlin, Richard, and Tao, Terence. 2010. The Kakeya set and maximal conjectures for algebraic varieties over finite fields. Mathematika, 56(1), 1–25.
Erdʺos, P. 1949. On a theorem of Hsu and Robbins. Ann. Math. Statistics, 20, 286–291.
Erdʺos, P. 1961. A problem about prime numbers and the random walk. II. Illinois J. Math., 5, 352–353.
Erdʺos, Paul. 1940. On the smoothness properties of a family of Bernoulli convolutions. Amer. J. Math., 62, 180–186.
Evans Steven, N. 1992. Polar and nonpolar sets for a tree indexed process. Ann. Probab., 20(2), 579–590.
Falconer, K.J. 1980. Continuity properties o. k-plane integrals and Besicovitch sets. Math. Proc. Cambridge Philos. Soc., 87(2), 221–226.
Falconer, K.J. 1982. Hausdorff dimension and the exceptional set of projections. Mathematika, 29(1), 109–115.
Falconer, K.J. 1985. The Geometry of Fractal Sets. Cambridge Tracts in Mathematics, vol. 85. Cambridge: Cambridge University Press.
Falconer, K.J. 1986. Sets with prescribed projections and Nikodým sets. Proc. London Math. Soc. (3), 53(1), 48–64.
Falconer, K.J. 1988. The Hausdorff dimension of self-affine fractals. Math. Proc. Cambridge Philos. Soc., 103(2), 339–350.
Falconer, K.J. 1989a. Dimensions and measures of quasi self-similar sets. Proc. Amer. Math. Soc., 106(2), 543–554.
Falconer, K.J. 1989b. Projections of random Cantor sets. J. Theoret. Probab., 2(1), 65–70.
Falconer, K.J. 1990. Fractal Geometry. Chichester: John Wiley & Sons Ltd.
Falconer, K.J. 2013. Dimensions of self-affine sets: a survey. Pages 115–134 of: Further Developments in Fractals and Related Fields. Trends Math. Birkhäuser/Springer, New York.
Falconer, K.J., and Howroyd, J.D. 1996. Projection theorems for box and packing dimensions. Math. Proc. Cambridge Philos. Soc., 119(2), 287–295.
Fang, X. 1990. The Cauchy integral of Calderόn and analytic capacity. Ph.D. thesis, Yale University.
Fefferman, Charles. 1971. The multiplier problem for the ball. Ann. of Math. (2), 94, 330–336.
Feller, William. 1966. An Introduction to Probability Theory and its Applications. Vol. II. New York: John Wiley & Sons Inc.
Ferguson, Andrew, Jordan, Thomas, and Shmerkin, Pablo. 2010. The Hausdorff dimension of the projections of self-affine carpets. Fund. Math., 209(3), 193–213.
Ferrari, Fausto, Franchi, Bruno, and Pajot, Hervé. 2007. The geometric traveling salesman problem in the Heisenberg group. Rev. Mat. Iberoam., 23(2), 437–480.
Folland, Gerald B. 1999. Real Analysis. Pure and Applied Mathematics (New York). New York: John Wiley & Sons Inc.
Ford Jr., L.R., and Fulkerson, D.R. 1962. Flows in Networks. Princeton, N.J.: Princeton University Press.
Freedman, David. 1971. Brownian Motion and Diffusion. San Francisco, Calif.: Holden-Day.
Freud, Geza. 1962. Über trigonometrische approximation und Fouriersche reihen. Math. Z., 78, 252–262.
Frostman, O. 1935. Potential d'equilibre et capacité des ensembles avec quelques applications `a la théorie des fonctions. Meddel. Lunds Univ. Math. Sen., 3(1-118).
Fujiwara, M., and Kakeya, S. 1917. On some problems for the maxima and minima for the curve of constant breadth and the in-revolvable curve of the equilateral triangle. Tohoku Math. J., 11, 92–110.
Furstenberg, H. 1967. Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Systems Theory, 1, 1–49.
Furstenberg, H. 1970. Intersections of Cantor sets and transversality of semigroups. Pages 41–59 of: Problems in Analysis (Sympos. Salomon Bochner, Princeton Univ., Princeton, N.J., 1969). Princeton Univ. Press, Princeton, N.J.
Furstenberg, H. 1981. Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton, N.J.: Princeton University Press.
Gabow, H.N. 1990. Data structures of weighted matching and nearest common ancestors with linking. In: Proceedings of the 1st Annual ACM–SIAM Symposium on Discrete Algorithms.
Garnett John, B. 1970. Positive length but zero analytic capacity. Proc. Amer. Math. Soc., 24, 696–699.
Garnett John, B. 1981. Bounded Analytic Functions. Academic Press.
Garnett John, B., and Marshall Donald, E. 2005. Harmonic Measure. New Mathematical Monographs, vol. 2. Cambridge: Cambridge University Press.
Gerver, Joseph. 1970. The differentiability of the Riemann function at certain rational multiples of?. Amer. J. Math., 92, 33–55.
Graczyk, Jacek, and Smirnov, Stanislav. 2009. Non-uniform hyperbolicity in complex dynamics. Invent. Math., 175(2), 335–415.
Graf, Siegfried, Mauldin R., Daniel, and Williams, S.C. 1988. The exact Hausdorff dimension in random recursive constructions. Mem. Amer. Math. Soc., 71(381), x+121.
Guth, Larry. 2010. The endpoint case of the Bennett–Carbery–Tao multilinear Kakeya conjecture. Acta Math., 205(2), 263–286.
Hahlomaa, Immo. 2005. Menger curvature and Lipschitz parametrizations in metric spaces. Fund. Math., 185(2), 143–169.
Hahlomaa, Immo. 2007. Curvature integral and Lipschitz parametrization in 1-regular metric spaces. Ann. Acad. Sci. Fenn. Math., 32(1), 99–123.
Hahlomaa, Immo. 2008. Menger curvature and rectifiability in metric spaces. Adv. Math., 219(6), 1894–1915.
Hamilton David, H. 1995. Length of Julia curves. Pacific J. Math., 169(1), 75–93.
Har-Peled, Sariel. 2011. Geometric Approximation Algorithms. Mathematical Surveys and Monographs, vol. 173. American Mathematical Society, Providence, RI.
Hardy, G.H. 1916. Weierstrass's non-differentiable function. Trans. Amer. Math. Soc., 17(3), 301–325.
Harris, T.E. 1960. A lower bound for the critical probability in a certain percolation process. Proc. Cambridge Philos. Soc., 56, 13–20.
Hartman, Philip, and Wintner, Aurel. 1941. On the law of the iterated logarithm. Amer. J. Math., 63, 169–176.
Hausdorff, Felix. 1918. Dimension und äußeres Maß. Math. Ann., 79(1-2), 157–179.
Hawkes, John. 1975. Some algebraic properties of small sets. Quart. J. Math. Oxford Ser. (2), 26(102), 195–201.
Hawkes, John. 1981. Trees generated by a simple branching process. J. London Math. Soc. (2), 24(2), 373–384.
Hedenmalm, Haken. 2015. Bloch functions, asymptotic variance and zero packing. preprint, arXiv:1602.03358 [math.CV].
Hewitt, Edwin, and Savage Leonard, J. 1955. Symmetric measures on Cartesian products. Trans. Amer. Math. Soc., 80, 470–501.
Hewitt, Edwin, and Stromberg, Karl. 1975. Real and Abstract Analysis. New York: Springer-Verlag.
Hochman, Michael. 2013. Dynamics on fractals and fractal distributions. arXiv:1008.3731v2.
Hochman, Michael. 2014. On self-similar sets with overlaps and inverse theorems for entropy. Ann. of Math. (2), 180(2), 773–822.
Hochman, Michael. 2015. On self-similar sets with overlaps and inverse theorems for entropy in Rd. In preparation.
Hochman, Michael, and Shmerkin, Pablo. 2012. Local entropy averages and projections of fractal measures. Ann. of Math. (2), 175(3), 1001–1059.
Housworth Elizabeth, Ann. 1994. Escape rate for 2-dimensional Brownian motion conditioned to be transient with application to Zygmund functions. Trans. Amer. Math. Soc., 343(2), 843–852.
Howroyd, J.D. 1995. On dimension and on the existence of sets of finite positive Hausdorff measure. Proc. London Math. Soc. (3), 70(3), 581–604.
Hsu, P.L., and Robbins, Herbert. 1947. Complete convergence and the law of large numbers. Proc. Nat. Acad. Sci. U.S.A., 33, 25–31.
Hunt Brian, R. 1994. The prevalence of continuous nowhere differentiable functions. Proc. Amer. Math. Soc., 122(3), 711–717.
Hunt Brian, R. 1998. The Hausdorff dimension of graphs of Weierstrass functions. Proc. Amer. Math. Soc., 126(3), 791–800.
Hunt Brian, R, Sauer, Tim, and Yorke James, A. 1992. Prevalence: a translationinvariant “almost every” on infinite-dimensional spaces. Bull. Amer. Math. Soc. (N.S.), 27(2), 217–238.
Hunt Brian, R, Sauer, Tim, and Yorke James, A. 1993. Prevalence. An addendum to: “Prevalence: a translation-invariant ‘almost every’ on infinite-dimensional spaces” [Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 2, 217–238; MR1161274 (93k:28018)]. Bull. Amer. Math. Soc. (N.S.), 28(2), 306–307.
Hunt, G.A. 1956. Some theorems concerning Brownian motion. Trans. Amer. Math. Soc., 81, 294–319.
Hutchinson John, E. 1981. Fractals and self-similarity. Indiana Univ. Math. J., 30(5), 713–747.
Ivrii, Oleg. 2016. Quasicircles of dimension 1 + k2 do not exist. Preprint, arXiv:1511.07240 [math.DS].
Izumi, Masako, Izumi, Shin-ichi, and Kahane, Jean-Pierre. 1965. Théor`emes élémentaires sur les séries de Fourier lacunaires. J. Analyse Math., 14, 235–246.
Jodeit Jr., Max. 1971. A note on Fourier multipliers. Proc. Amer. Math. Soc., 27, 423–424.
Jones Peter, W. 1989. Square functions, Cauchy integrals, analytic capacity, and harmonic measure. Pages 24–68 of: Harmonic Analysis and Partial Differential Equations (El Escorial, 1987). Lecture Notes in Math., vol. 1384. Berlin: Springer.
Jones Peter, W. 1990. Rectifiable sets and the traveling salesman problem. Invent. Math., 102(1), 1–15.
Jones Peter, W. 1991. The traveling salesman problem and harmonic analysis. Publ. Mat., 35(1), 259–267. Conference on Mathematical Analysis (El Escorial, 1989).
Jordan, Thomas, and Pollicott, Mark. 2006. Properties of measures supported on fat Sierpinski carpets. Ergodic Theory Dynam. Systems, 26(3), 739–754.
Joyce, H., and Preiss, D. 1995. On the existence of subsets of finite positive packing measure. Mathematika, 42(1), 15–24.
Juillet, Nicolas. 2010. A counterexample for the geometric traveling salesman problem in the Heisenberg group. Rev. Mat. Iberoam., 26(3), 1035–1056.
Kahane, J.-P. 1964. Lacunary Taylor and Fourier series. Bull. Amer. Math. Soc., 70, 199–213.
Kahane, J.-P. 1969. Trois notes sure les ensembles parfait linéaires. Enseigement Math., 15, 185–192.
Kahane, J.-P. 1985. Some Random Series of Functions. Second edn. Cambridge Studies in Advanced Mathematics, vol. 5. Cambridge: Cambridge University Press.
Kahane, J.-P., and Peyri`ere, J. 1976. Sur certaines martingales de Benoit Mandelbrot. Advances in Math., 22(2), 131–145.
Kahane, J.-P., Weiss, Mary, and Weiss, Guido. 1963. On lacunary power series. Ark. Mat., 5, 1–26 (1963).
Kakeya, S. 1917. Some problems on maxima and minima regarding ovals. Tohoku Science Reports, 6, 71–88.
Kakutani, Shizuo. 1944. Two-dimensional Brownian motion and harmonic functions. Proc. Imp. Acad. Tokyo, 20, 706–714.
Kamae, T., and Mend`es France, M. 1978. van der Corput's difference theorem. Israel J. Math., 31(3-4), 335–342.
Karatzas, Ioannis, and Shreve Steven, E. 1991. Brownian Motion and Stochastic Calculus. Second edn. Graduate Texts in Mathematics, vol. 113. New York: Springer-Verlag.
Karpińska, Boguslawa. 1999. Hausdorff dimension of the hairs without endpoints for exp z. C. R. Acad. Sci. Paris Sér. I Math., 328(11), 1039–1044.
Katz Nets, Hawk, and Tao, Terence. 2002. New bounds for Kakeya problems. J. Anal. Math., 87, 231–263. Dedicated to the memory of Thomas H. Wolff.
Katznelson, Y. 2001. Chromatic numbers of Cayley graphs on Z and recurrence. Combinatorica, 21(2), 211–219.
Kaufman, R. 1968. On Hausdorff dimension of projections. Mathematika, 15, 153–155.
Kaufman, R. 1969. Une propriété métrique du mouvement brownien. C. R. Acad. Sci. Paris Sér. A-B, 268, A727–A728.
Kaufman, R. 1972. Measures of Hausdorff-type, and Brownian motion. Mathematika, 19, 115–119.
Kechris Alexander, S. 1995. Classical Descriptive Set Theory. Graduate Texts in Mathematics, vol. 156. Springer-Verlag, New York.
Kenyon, Richard. 1997. Projecting the one-dimensional Sierpinski gasket. Israel J. Math., 97, 221–238.
Kenyon, Richard, and Peres, Yuval. 1991. Intersecting random translates of invariant Cantor sets. Invent. Math., 104(3), 601–629.
Kenyon, Richard, and Peres, Yuval. 1996. Hausdorff dimensions of sofic affineinvariant sets. Israel J. Math., 94, 157–178.
Khinchin, A.Y. 1924. Über einen Satz der Wahrscheinlichkeitrechnung. Fund. Mat., 6, 9–20.
Khinchin, A.Y. 1933. Asymptotische Gesetze der Wahrscheinlichkeitsrechnung. Springer-Verlag.
Khintchine, A. 1926. Über eine Klasse linearer diophantischer Approximationen. Rendiconti Circ. Math. Palermo, 50(2), 211–219.
Khoshnevisan, Davar. 1994. A discrete fractal in Z1 +. Proc. Amer. Math. Soc., 120(2), 577–584.
Kinney, J.R. 1968. A thin set of circles. Amer. Math. Monthly, 75(10), 1077–1081.
Knight Frank, B. 1981. Essentials of Brownian Motion and Diffusion. Mathematical Surveys, vol. 18. Providence, R.I.: American Mathematical Society.
Kochen, Simon, and Stone, Charles. 1964. A note on the Borel–Cantelli lemma. Illinois J. Math., 8, 248–251.
Kolmogorov, A. 1929. Über das Gesetz des iterierten Logarithmus. Mathematische Annalen, 101(1), 126–135.
Körner, T.W. 2003. Besicovitch via Baire. Studia Math., 158(1), 65–78.
Kruskal, Jr., Joseph, B. 1956. On the shortest spanning subtree of a graph and the traveling salesman problem. Proc. Amer. Math. Soc., 7, 48–50.
Kuipers, L., and Niederreiter, H. 1974. Uniform Distribution of Sequences. Wiley– Interscience [John Wiley & Sons], New York–London–Sydney. Pure and Applied Mathematics.
Łaba, Izabella. 2008. From harmonic analysis to arithmetic combinatorics. Bull. Amer. Math. Soc. (N.S.), 45(1), 77–115.
Lalley Steven, P., and Gatzouras, Dimitrios. 1992. Hausdorff and box dimensions of certain self-affine fractals. Indiana Univ. Math. J., 41(2), 533–568.
Lamperti, John. 1963. Wiener's test and Markov chains. J. Math. Anal. Appl., 6, 58–66.
Larman, D.H. 1967. On the Besicovitch dimension of the residual set of arbitrary packed disks in the plane. J. London Math. Soc., 42, 292–302.
Lawler Gregory, F. 1991. Intersections of Random Walks. Probability and its Applications. Boston, MA: Birkhäuser Boston Inc.
Lawler Gregory, F. 1996. The dimension of the frontier of planar Brownian motion. Electron. Comm. Probab., 1, no. 5, 29–47 (electronic).
Lawler Gregory, F., Schramm, Oded, and Werner, Wendelin. 2001a. The dimension of the planar Brownian frontier is 4/3. Math. Res. Lett., 8(4), 401–411.
Lawler Gregory, F., Schramm, Oded, and Werner, Wendelin. 2001b. Values of Brownian intersection exponents. I. Half-plane exponents. Acta Math., 187(2), 237–273.
Lawler Gregory, F., Schramm, Oded, and Werner, Wendelin. 2001c. Values of Brownian intersection exponents. II. Plane exponents. Acta Math., 187(2), 275–308.
Lawler Gregory, F., Schramm, Oded, and Werner, Wendelin. 2002. Values of Brownian intersection exponents. III. Two-sided exponents. Ann. Inst. H. Poincaré Probab. Statist., 38(1), 109–123.
Le Gall, J.-F. 1987. The exact Hausdorff measure of Brownian multiple points. Pages 107–137 of: Seminar on Stochastic Processes, 1986 (Charlottesville, VA., 1986). Progr. Probab. Statist., vol. 13. Birkhäuser Boston, Boston, MA.
Lehmann, E.L. 1966. Some concepts of dependence. Ann. Math. Statist., 37, 1137– 1153.
Lerman, Gilad. 2003. Quantifying curvelike structures of measures by using L2 Jones quantities. Comm. Pure Appl. Math., 56(9), 1294–1365.
Lévy, Paul. 1940. Le mouvement brownien plan. Amer. J. Math., 62, 487–550.
Lévy, Paul. 1948. Processus stochastiques et mouvement Brownien. Suivi d'une note de M. Lo`eve. Paris: Gauthier–Villars.
Lévy, Paul. 1953. La mesure de Hausdorff de la courbe du mouvement Brownien. Giorn. Ist. Ital. Attuari, 16, 1–37 (1954).
Lindenstrauss, Elon, and Varju, Peter P. 2014. Random walks in the group of Euclidean isometries and self-similar measures. arXiv:1405.4426.
Lyons, Russell. 1989. The Ising model and percolation on trees and tree-like graphs. Comm. Math. Phys., 125(2), 337–353.
Lyons, Russell. 1990. Random walks and percolation on trees. Ann. Probab., 18(3), 931–958.
Lyons, Russell. 1992. Random walks, capacity and percolation on trees. Ann. Probab., 20(4), 2043–2088.
Lyons, Russell, and Pemantle, Robin. 1992. Random walk in a random environment and first-passage percolation on trees. Ann. Probab., 20(1), 125–136.
Lyons, Russell, and Peres, Yuval. 2016. Probability on Trees and Networks. Cambridge University Press.
Makarov, N.G. 1989. Probability methods in the theory of conformal mappings. Algebra i Analiz, 1(1), 3–59.
Mandelbrot, Benoit. 1974. Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier. J. Fluid Mechanics, 62, 331–358.
Markowsky, Greg. 2011. On the expected exit time of planar Brownian motion from simply connected domains. Electron. Commun. Probab., 16, 652–663.
Marstrand, J.M. 1954. Some fundamental geometrical properties of plane sets of fractional dimensions. Proc. London Math. Soc. (3), 4, 257–302.
Marstrand, J.M. 1979. Packing planes in R3. Mathematika, 26(2), 180–183.(1980).
Mattila, Pertti. 1990. Orthogonal projections, Riesz capacities, and Minkowski content. Indiana Univ. Math. J., 39(1), 185–198.
Mattila, Pertti. 1995. Geometry of Sets and Measures in Euclidean Spaces. Cambridge Studies in Advanced Mathematics, vol. 44. Cambridge: Cambridge University Press. Fractals and rectifiability.
Mattila, Pertti. 2015. Fourier Analysis and Hausdorff Dimension. Cambridge Studies in Advanced Mathematics, vol. 150. Cambridge University Press.
Mattila, Pertti, and Mauldin, R. Daniel. 1997. Measure and dimension functions: measurability and densities. Math. Proc. Cambridge Philos. Soc., 121(1), 81–100.
Mattila, Pertti, and Vuorinen, Matti. 1990. Linear approximation property, Minkowski dimension, and quasiconformal spheres. J. London Math. Soc. (2), 42(2), 249– 266.
Mattila, Pertti, Melnikov Mark, S., and Verdera, Joan. 1996. The Cauchy integral, analytic capacity, and uniform rectifiability. Ann. of Math. (2), 144(1), 127–136.
Mauldin R., Daniel, and Williams, S.C. 1986. On the Hausdorff dimension of some graphs. Trans. Amer. Math. Soc., 298(2), 793–803.
McKean Jr. Henry, P. 1961. A problem about prime numbers and the random walk. I. Illinois J. Math., 5, 351.
McKean Jr. Henry, P. 1955. Hausdorff–Besicovitch dimension of Brownian motion paths. Duke Math. J., 22, 229–234.
McMullen Curtis, T. 1984. The Hausdorff dimension of general Sierpiński carpets. Nagoya Math. J., 96, 1–9.
McMullen Curtis, T. 1998. Hausdorff dimension and conformal dynamics. III. Computation of dimension. Amer. J. Math., 120(4), 691–721.
Melnikov Mark, S., and Verdera, Joan. 1995. A geometric proof of the L2 boundedness of the Cauchy integral on Lipschitz graphs. Internat. Math. Res. Notices, 325–331.
Milnor, John. 2006. Dynamics in One Complex Variable. Third edn. Annals of Mathematics Studies, vol. 160. Princeton, NJ: Princeton University Press.
Mitchell Joseph, S.B. 1999. Guillotine subdivisions approximate polygonal subdivisions: a simple polynomial-time approximation scheme for geometric TSP, k- MST, and related problems. SIAM J. Comput., 28(4), 1298–1309.(electronic).
Mitchell Joseph, S.B. 2004. Shortest paths and networks. Chap. 27, pages 607–641 of: Goodman, Jacob E., and O'Rourke, Joseph (eds), Handbook of Discrete and Computational Geometry (2nd Edition). Boca Raton, FL: Chapman & Hall/CRC.
Montgomery Hugh, L. 2001. Harmonic analysis as found in analytic number theory. Pages 271–293 of: Twentieth Century Harmonic Analysis – a Celebration (Il Ciocco, 2000). NATO Sci. Ser. II Math. Phys. Chem., vol. 33. Kluwer Acad. Publ., Dordrecht.
Moran, P.A.P. 1946. Additive functions of intervals and Hausdorff measure. Proc. Cambridge Philos. Soc., 42, 15–23.
Mörters, Peter, and Peres, Yuval. 2010. Brownian Motion. With an appendix by Oded Schramm and Wendelin Werner. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge: Cambridge University Press.
Muller Mervin, E. 1956. Some continuous Monte Carlo methods for the Dirichlet problem. Ann. Math. Statist., 27, 569–589.
Nazarov, F., Peres, Y., and Volberg, A. 2010. The power law for the Buffon needle probability of the four-corner Cantor set. Algebra i Analiz, 22(1), 82–97.
Nevanlinna, Rolf. 1936. Eindeutige Analytische Funktionen. Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, vol. 46. J. Springer, Berlin.
Newhouse Sheldon, E. 1970. Nondensity of axiom A(a) on S2. Pages 191–202 of: Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968). Amer. Math. Soc., Providence, R.I.
Nikodym, O. 1927. Sur la measure des ensembles plan dont tous les points sont rectalineair ément accessibles. Fund. Math., 10, 116–168.
Oberlin, Richard. 2010. Two bounds for the X-ray transform. Math. Z., 266(3), 623–644.
Oh, Hee. 2014. Apollonian circle packings: dynamics and number theory. Jpn. J. Math., 9(1), 69–97.
Okikiolu, Kate. 1992. Characterization of subsets of rectifiable curves in R n. J. London Math. Soc. (2), 46(2), 336–348.
O'Neill Michael, D. 2000. Anderson's conjecture for domains with fractal boundary. Rocky Mountain J. Math., 30(1), 341–352.
Pajot, Hervé. 2002. Analytic Capacity, Rectifiability, Menger Curvature and the Cauchy Integral. Lecture Notes in Mathematics, vol. 1799. Berlin: Springer-Verlag.
Pál, Julius. 1921. Ein minimumproblem für ovale. Math. Ann., 83(3-4), 311–319.
Paley, R.E.A.C., Wiener, N., and Zygmund, A. 1933. Notes on random functions. Math. Z., 37(1), 647–668.
Parry, William. 1964. Intrinsic Markov chains. Trans. Amer. Math. Soc., 112, 55–66.
Pemantle, Robin. 1997. The probability that Brownian motion almost contains a line. Ann. Inst. H. Poincaré Probab. Statist., 33(2), 147–165.
Pemantle, Robin, and Peres, Yuval. 1994. Domination between trees and application to an explosion problem. Ann. Probab., 22(1), 180–194.
Pemantle, Robin, and Peres, Yuval. 1995. Galton–Watson trees with the same mean have the same polar sets. Ann. Probab., 23(3), 1102–1124.
Peres, Yuval. 1994a. The packing measure of self-affine carpets. Math. Proc. Cambridge Philos. Soc., 115(3), 437–450.
Peres, Yuval. 1994b. The self-affine carpets of McMullen and Bedford have infinite Hausdorff measure. Math. Proc. Cambridge Philos. Soc., 116(3), 513–526.
Peres, Yuval. 1996a. Points of increase for random walks. Israel J. Math., 95, 341–347.
Peres, Yuval. 1996b. Remarks on intersection-equivalence and capacity-equivalence. Ann. Inst. H. Poincaré Phys. Théor., 64(3), 339–347.
Peres, Yuval, and Schlag, Wilhelm. 2000. Smoothness of projections, Bernoulli convolutions, and the dimension of exceptions. Duke Math. J., 102(2), 193–251.
Peres, Yuval, and Schlag, Wilhelm. 2010. Two Erdʺos problems on lacunary sequences: chromatic number and Diophantine approximation. Bull. Lond. Math. Soc., 42(2), 295–300.
Peres, Yuval, and Shmerkin, Pablo. 2009. Resonance between Cantor sets. Ergodic Theory Dynam. Systems, 29(1), 201–221.
Peres, Yuval, and Solomyak, Boris. 1996. Absolute continuity of Bernoulli convolutions, a simple proof. Math. Res. Lett., 3(2), 231–239.
Peres, Yuval, and Solomyak, Boris. 2000. Problems on self-similar sets and self-affine sets: an update. Pages 95–106 of: Fractal Geometry and Stochastics, II (Greifswald/Koserow, 1998). Progr. Probab., vol. 46. Birkhäuser, Basel.
Peres, Yuval, and Solomyak, Boris. 2002. How likely is Buffon's needle to fall near a planar Cantor set. Pacific J. Math., 204(2), 473–496.
Perron, Oskar. 1929. Ü ber stabilität und asymptotisches verhalten der integrale von differentialgleichungssystemen. Math. Z., 29(1), 129–160.
Prause, István, and Smirnov, Stanislav. 2011. Quasisymmetric distortion spectrum. Bull. Lond. Math. Soc., 43(2), 267–277.
Przytycki, F., and Urbański, M. 1989. On the Hausdorff dimension of some fractal sets. Studia Math., 93(2), 155–186.
Quilodrán, René. 2009/10. The joints problem in Rn. SIAM J. Discrete Math., 23(4), 2211–2213.
Ray, Daniel. 1963. Sojourn times and the exact Hausdorff measure of the sample path for planar Brownian motion. Trans. Amer. Math. Soc., 106, 436–444.
Revuz, Daniel, and Yor, Marc. 1994. Continuous Martingales and Brownian motion. Second edn. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293. Berlin: Springer-Verlag.
Rippon, P.J., and Stallard, G.M. 2005. Dimensions of Julia sets of meromorphic functions. J. London Math. Soc. (2), 71(3), 669–683.
Rogers, C.A., and Taylor, S.J. 1959. The analysis of additive set functions in Euclidean space. Acta Math., 101, 273–302.
Rohde, S. 1991. On conformal welding and quasicircles. Michigan Math. J., 38(1), 111–116.
Root, D.H. 1969. The existence of certain stopping times on Brownian motion. Ann. Math. Statist., 40, 715–718.
Rudin, Walter. 1987. Real and Complex Analysis. Third edn. New York: McGraw-Hill Book Co.
Ruzsa, I.Z., and Székely, G.J. 1982. Intersections of traces of random walks with fixed sets. Ann. Probab., 10(1), 132–136.
Saint Raymond, Xavier, and Tricot, Claude. 1988. Packing regularity of sets in n-space. Math. Proc. Cambridge Philos. Soc., 103(1), 133–145.
Salem, R., and Zygmund, A. 1945. Lacunary power series and Peano curves. Duke Math. J., 12, 569–578.
Saraf, Shubhangi, and Sudan, Madhu. 2008. An improved lower bound on the size of Kakeya sets over finite fields. Anal. PDE, 1(3), 375–379.
Sawyer, Eric. 1987. Families of plane curves having translates in a set of measure zero. Mathematika, 34(1), 69–76.
Schief, Andreas. 1994. Separation properties for self-similar sets. Proc. Amer. Math. Soc., 122(1), 111–115.
Schleicher, Dierk. 2007. The dynamical fine structure of iterated cosine maps and a dimension paradox. Duke Math. J., 136(2), 343–356.
Schoenberg, I.J. 1962. On the Besicovitch–Perron solution of the Kakeya problem. Pages 359–363 of: Studies in Mathematical Analysis and Related Topics. Stanford, Calif.: Stanford Univ. Press.
Schul, Raanan. 2005. Subsets of rectifiable curves in Hilbert space – the analyst's TSP. Ph.D. thesis, Yale University.
Schul, Raanan. 2007a. Ahlfors-regular curves in metric spaces. Ann. Acad. Sci. Fenn. Math., 32(2), 437–460.
Schul, Raanan. 2007b. Analyst's traveling salesman theorems. A survey. Pages 209–220 of: In the Tradition of Ahlfors–Bers. IV. Contemp. Math., vol. 432. Providence, RI: Amer. Math. Soc.
Schul, Raanan. 2007c. Subsets of rectifiable curves in Hilbert space – the analyst's TSP. J. Anal. Math., 103, 331–375.
Schwartz, J.T. 1980. Fast probabilistic algorithms for verification of polynomial identities. J. Assoc. Comput. Mach., 27(4), 701–717.
Selberg, Atle. 1952. The general sieve-method and its place in prime number theory. Pages 286–292 of: Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950, vol. 1. Providence, R. I.: Amer. Math. Soc.
Shannon, C.E. 1948. A mathematical theory of communication. Bell System Tech. J., 27, 379–423, 623–656.
Shen, Weixiao. 2015. Hausdorff dimension of the graphs of the classical Weierstrass functions. Preprint, arXiv:1505.03986 [math.DS].
Shmerkin, Pablo. 2014. On the exceptional set for absolute continuity of Bernoulli convolutions. Geom. Funct. Anal., 24(3), 946–958.
Shmerkin, Pablo, and Solomyak, Boris. 2014. Absolute continuity of self-similar measures, their projections and convolutions. arXiv:1406.0204.
Skorokhod, A.V. 1965. Studies in the Theory of Random Processes. Translated from the Russian by Scripta Technica, Inc. Addison-Wesley Publishing Co., Inc., Reading, Mass.
Smirnov, Stanislav. 2010. Dimension of quasicircles. Acta Math., 205(1), 189–197.
Solecki, Sławomir. 1996. On Haar null sets. Fund. Math., 149(3), 205–210.
Solomyak, Boris. 1995. On the random series © ± n(an Erdʺos problem). Ann. of Math. (2), 142(3), 611–625.
Solomyak, Boris. 1997. On the measure of arithmetic sums of Cantor sets. Indag. Math. (N.S.), 8(1), 133–141.
Solomyak, Boris, and Xu, Hui. 2003. On the ‘Mandelbrot set’ for a pair of linear maps and complex Bernoulli convolutions. Nonlinearity, 16(5), 1733–1749.
Spitzer, Frank. 1964. Principles of Random Walk. The University Series in Higher Mathematics. D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London.
Strassen, V. 1964. An invariance principle for the law of the iterated logarithm. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 3, 211–226 (1964).
Stratmann Bernd, O. 2004. The exponent of convergence of Kleinian groups; on a theorem of Bishop and Jones. Pages 93–107 of: Fractal Geometry and Stochastics III. Progr. Probab., vol. 57. Basel: Birkhäuser.
Sullivan, Dennis. 1982. Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics. Acta Math., 149(3-4), 215–237.
Sullivan, Dennis. 1984. Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups. Acta Math., 153(3-4), 259–277.
Talagrand, Michel, and Xiao, Yimin. 1996. Fractional Brownian motion and packing dimension. J. Theoret. Probab., 9(3), 579–593.
Taylor S., James. 1953. The Hausdorff 〈-dimensional measure of Brownian paths in n-space. Proc. Cambridge Philos. Soc., 49, 31–39.
Taylor S., James. 1964. The exact Hausdorff measure of the sample path for planar Brownian motion. Proc. Cambridge Philos. Soc., 60, 253–258.
Taylor S., James. 1966. Multiple points for the sample paths of the symmetric stable process. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 5, 247–264.
Taylor S., James, and Tricot, Claude. 1985. Packing measure, and its evaluation for a Brownian path. Trans. Amer. Math. Soc., 288(2), 679–699.
Taylor S., James, and Wendel, J.G. 1966. The exact Hausdorff measure of the zero set of a stable process. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 6, 170–180.
Tolsa, Xavier. 2003. Painlevé's problem and the semiadditivity of analytic capacity. Acta Math., 190(1), 105–149.
Tolsa, Xavier. 2014. Analytic Capacity, the Cauchy Transform, and Nonhomogeneous Calderόn–Zygmund theory. Progress in Mathematics, vol. 307. Birkhäuser/Springer, Cham.
Tricot, Claude. 1981. Douze definitions de la densité logarithmique. Comptes Rendus Acad. Sci. Paris, 293, 549–552.
Tricot, Claude. 1984. A new proof for the residual set dimension of the Apollonian packing. Math. Proc. Cambridge Philos. Soc., 96(3), 413–423.
Tricot, Claude. 1982. Two definitions of fractional dimension. Math. Proc. Cambridge Philos. Soc., 91(1), 57–74.
Tukia, Pekka. 1989. Hausdorff dimension and quasisymmetric mappings. Math. Scand., 65(1), 152–160.
Urbański, Mariusz. 1990. The Hausdorff dimension of the graphs of continuous selfaffine functions. Proc. Amer. Math. Soc., 108(4), 921–930.
Urbański, Mariusz. 1991. On the Hausdorff dimension of a Julia set with a rationally indifferent periodic point. Studia Math., 97(3), 167–188.
Urbański, Mariusz. 1997. Geometry and ergodic theory of conformal non-recurrent dynamics. Ergodic Theory Dynam. Systems, 17(6), 1449–1476.
van der Corput, J.G. 1931. Diophantische ungleichungen. I. Zur gleichverteilung modulo eins. Acta Math., 56(1), 373–456.
Volkmann, Bodo. 1958. Ü ber Hausdorffsche Dimensionen von Mengen, die durch Zifferneigenschaften charakterisiert sind. VI. Math. Z., 68, 439–449.
Weierstrass, K. 1872. Ü ber contiuirliche functionen eines reellen arguments, die für keinen werth des letzeren einen bestimmten differentialquotienten besitzen. Königl. Akad. Wiss., 3, 71–74. Mathematische Werke II.
Wiener, N. 1923. Differential space. J. Math. Phys., 2(6), 1319–1362.
Wolff, Thomas. 1995. An improved bound for Kakeya type maximal functions. Rev. Mat. Iberoamericana, 11(3), 651–674.
Wolff, Thomas. 1999. Recent work connected with the Kakeya problem. Pages 129–162 of: Prospects in Mathematics (Princeton, NJ, 1996). Providence, RI: Amer. Math. Soc.
Xiao, Yimin. 1996. Packing dimension, Hausdorff dimension and Cartesian product sets. Math. Proc. Cambridge Philos. Soc., 120(3), 535–546.
Zdunik, Anna. 1990. Parabolic orbifolds and the dimension of the maximal measure for rational maps. Invent. Math., 99(3), 627–649.
Zhan, Dapeng. 2011. Loop-erasure of planar Brownian motion. Comm. Math. Phys., 303(3), 709–720.
Zippel, Richard. 1979. Probabilistic algorithms for sparse polynomials. Pages 216– 226 of: Symbolic and Algebraic Computation (EUROSAM ‘79, Internat. Sympos., Marseille, 1979). Lecture Notes in Comput. Sci., vol. 72. Berlin: Springer.
Zygmund, A. 1959. Trigonometric Series. Cambridge University Press.

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Book summary page views

Total views: 0 *
Loading metrics...

* Views captured on Cambridge Core between #date#. This data will be updated every 24 hours.

Usage data cannot currently be displayed.